Higher_Order_Rewriting_Union_Beta 2019-03-28 22.10 pair #432270349

details

property	value
status	complete
benchmark	pical.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n050.star.cs.uiowa.edu
space	Hamana_17

run statistics

property	value
solver	Wanda 2.1c
configuration	default
runtime (wallclock)	0.069215 seconds
cpu usage	0.069256
user time	0.061545
system time	0.007711
max virtual memory	113176.0
max residence set size	3336.0

stage attributes

key	value
starexec-result	YES

output

0.00/0.05	YES
0.00/0.06	We consider the system theBenchmark.
0.00/0.06	
0.00/0.06	  Alphabet:
0.00/0.06	
0.00/0.06	    NIL : [] --> A 
0.00/0.06	    in : [] --> N -> (N -> A) -> A 
0.00/0.06	    new : [] --> (N -> A) -> A 
0.00/0.06	    out : [] --> N -> N -> A -> A 
0.00/0.06	    sum : [] --> A -> A -> A 
0.00/0.06	    tau : [] --> A -> A 
0.00/0.06	
0.00/0.06	  Rules:
0.00/0.06	
0.00/0.06	    sum NIL x => x 
0.00/0.06	    new (/\x.y) => y 
0.00/0.06	    new (/\x.sum (f x) (g x)) => sum (new (/\y.f y)) (new (/\z.g z)) 
0.00/0.06	    new (/\x.out x y (f x)) => NIL 
0.00/0.06	    new (/\x.out y z (f x)) => out y z (new (/\u.f u)) 
0.00/0.06	    new (/\x.in y (/\z.f x z)) => in y (/\u.new (/\v.f v u)) 
0.00/0.06	    new (/\x.tau (f x)) => tau (new (/\y.f y)) 
0.00/0.06	    new (/\x.in x (/\y.f x y)) => NIL 
0.00/0.06	
0.00/0.06	Using the transformations described in [Kop11], this system can be brought in a form without leading free variables in the left-hand side, and where the left-hand side of a variable is always a functional term or application headed by a functional term.
0.00/0.06	
0.00/0.06	We now transform the resulting AFS into an AFSM by replacing all free variables by meta-variables (with arity 0).  This leads to the following AFSM:
0.00/0.06	
0.00/0.06	  Alphabet:
0.00/0.06	
0.00/0.06	    NIL : [] --> A 
0.00/0.06	    in : [N * N -> A] --> A 
0.00/0.06	    new : [N -> A] --> A 
0.00/0.06	    out : [N * N * A] --> A 
0.00/0.06	    sum : [A * A] --> A 
0.00/0.06	    tau : [A] --> A 
0.00/0.06	    ~AP1 : [N -> A * N] --> A 
0.00/0.06	    ~AP2 : [N -> N -> A * N] --> N -> A 
0.00/0.06	
0.00/0.06	  Rules:
0.00/0.06	
0.00/0.06	    sum(NIL, X) => X 
0.00/0.06	    new(/\x.X) => X 
0.00/0.06	    new(/\x.sum(~AP1(F, x), ~AP1(G, x))) => sum(new(/\y.~AP1(F, y)), new(/\z.~AP1(G, z))) 
0.00/0.06	    new(/\x.out(x, X, ~AP1(F, x))) => NIL 
0.00/0.06	    new(/\x.out(X, Y, ~AP1(F, x))) => out(X, Y, new(/\y.~AP1(F, y))) 
0.00/0.06	    new(/\x.in(X, /\y.~AP1(~AP2(F, x), y))) => in(X, /\z.new(/\u.~AP1(~AP2(F, u), z))) 
0.00/0.06	    new(/\x.tau(~AP1(F, x))) => tau(new(/\y.~AP1(F, y))) 
0.00/0.06	    new(/\x.in(x, /\y.~AP1(~AP2(F, x), y))) => NIL 
0.00/0.06	    ~AP1(F, X) => F X 
0.00/0.06	    ~AP2(F, X) => F X 
0.00/0.06	
0.00/0.06	Symbol ~AP2 is an encoding for application that is only used in innocuous ways.  We can simplify the program (without losing non-termination) by removing it.  This gives:
0.00/0.06	
0.00/0.06	  Alphabet:
0.00/0.06	
0.00/0.06	    NIL : [] --> A 
0.00/0.06	    in : [N * N -> A] --> A 
0.00/0.06	    new : [N -> A] --> A 
0.00/0.06	    out : [N * N * A] --> A 
0.00/0.06	    sum : [A * A] --> A 
0.00/0.06	    tau : [A] --> A 
0.00/0.06	    ~AP1 : [N -> A * N] --> A 
0.00/0.06	
0.00/0.06	  Rules:
0.00/0.06	
0.00/0.06	    sum(NIL, X) => X 
0.00/0.06	    new(/\x.X) => X 
0.00/0.06	    new(/\x.sum(~AP1(F, x), ~AP1(G, x))) => sum(new(/\y.~AP1(F, y)), new(/\z.~AP1(G, z))) 
0.00/0.06	    new(/\x.out(x, X, ~AP1(F, x))) => NIL 
0.00/0.06	    new(/\x.out(X, Y, ~AP1(F, x))) => out(X, Y, new(/\y.~AP1(F, y))) 
0.00/0.06	    new(/\x.in(X, /\y.~AP1(F(x), y))) => in(X, /\z.new(/\u.~AP1(F(u), z))) 
0.00/0.06	    new(/\x.tau(~AP1(F, x))) => tau(new(/\y.~AP1(F, y))) 
0.00/0.06	    new(/\x.in(x, /\y.~AP1(F(x), y))) => NIL 
0.00/0.06	    ~AP1(F, X) => F X 
0.00/0.06	
0.00/0.06	We use rule removal, following [Kop12, Theorem 2.23].
0.00/0.06	
0.00/0.06	This gives the following requirements (possibly using Theorems 2.25 and 2.26 in [Kop12]):
0.00/0.06	
0.00/0.06	  sum(NIL, X) >? X 
0.00/0.06	  new(/\x.X) >? X 
0.00/0.06	  new(/\x.sum(~AP1(F, x), ~AP1(G, x))) >? sum(new(/\y.~AP1(F, y)), new(/\z.~AP1(G, z))) 
0.00/0.06	  new(/\x.out(x, X, ~AP1(F, x))) >? NIL 
0.00/0.06	  new(/\x.out(X, Y, ~AP1(F, x))) >? out(X, Y, new(/\y.~AP1(F, y))) 
0.00/0.06	  new(/\x.in(X, /\y.~AP1(F(x), y))) >? in(X, /\z.new(/\u.~AP1(F(u), z))) 
0.00/0.06	  new(/\x.tau(~AP1(F, x))) >? tau(new(/\y.~AP1(F, y))) 
0.00/0.06	  new(/\x.in(x, /\y.~AP1(F(x), y))) >? NIL 
0.00/0.06	  ~AP1(F, X) >? F X 
0.00/0.06	
0.00/0.06	We orient these requirements with a polynomial interpretation in the natural numbers.
0.00/0.06	
0.00/0.06	The following interpretation satisfies the requirements:
0.00/0.06	
0.00/0.06	  NIL = 0 
0.00/0.06	  in = \y0G1.3 + y0 + 2G1(0) 
0.00/0.06	  new = \G0.2 + 3G0(0) 
0.00/0.06	  out = \y0y1y2.3 + y0 + y1 + 2y2 
0.00/0.06	  sum = \y0y1.3 + y0 + y1 
0.00/0.06	  tau = \y0.3 + y0 
0.00/0.06	  ~AP1 = \G0y1.3 + y1 + G0(y1)

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